The Lowland League leaders, who have not conceded a goal for 621 minutes, netted the opener midway through the first half at New Dundas Park as former Musselburgh Athletic frontman Nathan Evans converted a Neil Martyniuk corner.
Eight minutes later Bob Wilson's accurate finish from a Lee Currie corner doubled Rose's advantage.
It was game over for the Borders club as Evans added a second and the home side's third nine minutes before the half-time interval.
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The only goal of the second half arrived on the hour, a fantastic finish from George Hunter from the edge of the penalty area.
Rose boss Robbie Horn said afterwards: "It wasn't a brilliant performance but we did what we had to do. We controlled the game throughout and limited them to very few chances. They had an opportunity in the second half where Mark (Weir) has made a save and that was it.
"But it was just a wee bit too slow or pedestrian at times. We kept a clean sheet and scored a few goals so we can be fairly happy with that and we can now look forward to the game against Alloa on Saturday."
Ahead of this weekend's Scottish Cup third-round tie away to the League One side, Horn added: "We've got quite a few injuries at this moment in time so we'll just have to see who is fit at training this week.
“We'll prepare properly, it's a big game for the club and one we're really looking forward to. It's a free hit and if we can perform to the levels we can and you get a bit of luck on the day, you just never know."
Elsewhere in the South Challenge Cup, Civil Service Strollers are also through following their 3-1 victory over Threave Rovers at Palmerston.
Matthew Shaw, Robbie Cole and Joao Balde were all on target for Gary Jardine's side at the home of Queen of the South.
However, there was disappointment for Edinburgh University as they crashed out 3-1 to Vale of Leven of the West of Scotland League.